Question: Simplify the following expression and state the condition under which the simplification is valid: $x = \dfrac{t^2 + t - 30}{t^2 - 5t}$
First factor the expressions in the numerator and denominator. $ \dfrac{t^2 + t - 30}{t^2 - 5t} = \dfrac{(t + 6)(t - 5)}{(t)(t - 5)} $ Notice that the term $(t - 5)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(t - 5)$ gives: $x = \dfrac{t + 6}{t}$ Since we divided by $(t - 5)$, $t \neq 5$. $x = \dfrac{t + 6}{t}; \space t \neq 5$